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Based on one observation X from a N(θ, θ) distribution where θ(> 0) is an unknown parameter, this article provides an exact expression of the minimum variance unbiased estimator of θ based on the complete and sufficient statistic ∣X∣. Even though N(θ, θ) belongs to a one-parameter exponential family, I discuss some small surprises and intricacies that are worth noting. [ABSTRACT FROM AUTHOR]

Restricted maximum likelihood (REML) estimation of the parameters of the mixed model has become commonplace, even becoming the default option in many statistical software packages. However, a review of the literature indicates a need to update and clarify model selection techniques under REML, as ambiguities exist on the appropriateness of existing information criteria in this setting. A simulation study as well as an application assisted in gaining an understanding of the performance of information criteria in selecting the best model when using REML estimation. [ABSTRACT FROM AUTHOR]